reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  X1,X2 are_separated iff ex Y1, Y2 being closed non empty SubSpace of X
  st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  thus X1,X2 are_separated implies ex Y1, Y2 being closed non empty SubSpace
of X st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses
  X1
  proof
    assume X1,X2 are_separated;
    then A1,A2 are_separated;
    then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: C1 misses A2 & C2 misses A1 and
A4: C1 is closed and
A5: C2 is closed by Th42;
    C1 is non empty by A1,XBOOLE_1:3;
    then consider Y1 being strict closed non empty SubSpace of X such that
A6: C1 = the carrier of Y1 by A4,Th15;
    C2 is non empty by A2,XBOOLE_1:3;
    then consider Y2 being strict closed non empty SubSpace of X such that
A7: C2 = the carrier of Y2 by A5,Th15;
    take Y1,Y2;
    thus thesis by A1,A2,A3,A6,A7,Th4;
  end;
  given Y1, Y2 being closed non empty SubSpace of X such that
A8: X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1;
  now
    let A1, A2 be Subset of X such that
A9: A1 = the carrier of X1 & A2 = the carrier of X2;
    ex C1, C2 being Subset of X st A1 c= C1 & A2 c= C2 & C1 misses A2 &
    C2 misses A1 & C1 is closed & C2 is closed
    proof
      reconsider C2 = the carrier of Y2 as Subset of X by Th1;
      reconsider C1 = the carrier of Y1 as Subset of X by Th1;
      take C1,C2;
      thus thesis by A8,A9,Th4,Th11;
    end;
    hence A1,A2 are_separated by Th42;
  end;
  hence thesis;
end;
